In many applications the viscous terms become only important in parts of the computational domain. As a typical example serves the flow around the wing of an airplane, where close to the wing the viscous terms in the Navier Stokes equations are essential for the solution, while away from the wing, Euler's equations would suffice for the simulation. This leads to the interesting problem of finding coupling conditions between these two partial differential equations of different type. While coupling conditions have been developed in the literature, for example by using a limiting procedure on a globally viscous problem, we are interested here to develop coupling conditions which lead to coupled solutions which are as close as possible to the fully viscous solution. We develop our new approach on the one dimensional model problem of advection reaction diffusion equations with pure advection reaction approximation in subregions, which leads to the problem of coupling first and second order operators. Our guiding principle for finding transmission conditions is an operator factorization, and we show both analytically and numerically that the new coupling conditions lead to coupled solutions which are much closer to the fully viscous ones than other coupling conditions from the literature.